Can You Solve The 4 Hats Logic Puzzle?
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- čas přidán 18. 10. 2023
- Four friends claim they have been wrongly imprisoned. They are not criminals but mere harmless logicians! The warden says, "If you are in fact logicians I will set you free. But first you have to prove it to me." Can they work it out?
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If Logician 1 sees two hats the same colour then he knows his hat is the opposite colour and speaks out. After a while of not speaking out Logician 2 realises that 1 must be looking at two different colours and therefore his hat is the opposite colour to the one in front of him and he speaks out. Basically it's the same, but without the need for the logic table.
Yea the table just made things more complicates
Agreed, no need to work through all permutations.
Absolutely; this was very easy to figure out without complicating the problem with all the possibilities. Once #1 doesn’t immediately answer, #2 knows his hat is not the same as #3.
This is how I solved it too.
In seconds nice thinking 👍
Number 1 isn't confused, he's a perfect logician, he's impatiently waiting for 2 to hurry up and get the message.
Top comment
number 2 never got the message. He's trained in logic alone, and ends up never knowing if number 1 doens't answer because he didn't figure it out, doesn't want to, or any other reason
But the weren't supposed to talk to each other.
@@larrydene6379they didnt talk
they're friends and in jail. 2 knows 1 is trying.@@shiinondogewalker2809
Three logicians walk into a bar. The bartender asks them "Would you three like something to drink?" The first logician says "All three of us? I don't know." The second logician says "I don't know either." The third says "Yes, we would."
The version I read went:
"Can I get you all a beer?"
"I don't know"
"I don't know"
"Yes"
I prefer it more than yours tbh, less obvious
@@Blade.5786nah, your Version is worded badly. There is no way for them to logically deduce whether he *can* get them a beer. Might be considered nitpicking but seems like a relevant distinction in a logic joke which is mainly based around wording. This would be a better way to phrase it:
Do you all want a beer?
Idk
Idk
Yes
i wont get it
@@SH0907 Actually there is, the logical deduction would be that he can (considering the fact that he's the bartender). Using logic even further, this is information the bartender himself is aware of, and therefore wouldn't be asking if he wasn't implying something else (whether they *wanted* a beer). Logic is ingrained in our everyday speech, so it's not exactly wrong to use it in a logic joke.
@@antijokecommenter5003
1's answer indicates 1 wants a beer but does not know about 2 and 3.
2 and 3 then know 1 wants a beer.
2's answer indicates 2 wants a beer but still does not know about 3.
3 then knows 1 and 2 want a beer.
Since 3 wants a beer, 3 now knows all want a beer.
You don't really need to enumerate all 6 possibilities in this case. If the 2nd and 3rd person wears the same color, then the 1st person can immediately answer. If they wear different color, then the 1st will stay silent. And that silence becomes information to the 2nd person, who knows the color of the 3rd and, hence, also knows his. I think this concept is called common knowledge
That is my thinking also.
This is how I worked it out as well 👍
came here to write this as well
Even more - after the second man claim his hat, third one can do it as well
exactly
...I feel like you've made this more complicated than it had to be.
Video had to be long enough to be monetized
@@Nako3 isnt it min 8 minutes for monetization tho
@@randomgamer-te8op Im not sure
@@Nako3then why are writting such a bs?!
complicated? how???
It could be expressed more simply. If #1 sees two of the same color hat, he says he has the other one. If #1 says nothing, then #2 knows he and the guy in front of him have different color hats, so he says the opposite of whatever color he sees on the guy in front of him.
Unfortunately not long enough an explanation to make this video profitable
Still not 100% sure. Number one could be bad at his job.
yes this also works!! there are so many other possibilities to solve this puzzle.
Part of the rules of the puzzle is that all four people are PERFECT logicians.@@mattwatson7510
Absolutely. The author again is supposedly making all three friends equally capable, with the same skills.. that's not true.
I misheard the rules. I thought all four had to state the color of their own hat for them to be set free, and I just couldn't see that happening.
Same. In that case only 2 and 3 would know their color.
me too
That would've been a far more interesting puzzle.
In the case of 2&3 having same colour A, 1 will shout colour B. 2 can then conclude he is wearing colour A (because 1 sees two same hats). 3 can conclude he is wearing colour A on same premise. 4 knows he's wearing colour B by process of elimination.
In the case of 2&3 having different colours (A&B, respectively), 1 will stay quiet and 2 will conclude he is wearing A. 1 still doesn't know if he's A or B, and stays quiet. 3 can now conclude this is a "different colour scenario", and knows he is B (opposite of 2). I see no way for 1&4 to figure out what they're wearing from here.
@@BlurbFish Yep, they'd have a 50/50 chance, which is what they all start with anyway...
Once 2 has spoken, 3 knows their hat is red (if it had been the same as 2, 1 would have spoken first; so, knowing now that 2 is blue, 3 knows theirs must be the opposite). But here's where it stalls - no-one can see either 1's or 4's hat. They all know they're opposites, but not which way round they are. It could be resolved if 3 were allowed to turn round when they'd got their answer, but that's kinda kludgy. It would be nice if we could find a way for 4 to be the one that resolves it - say, if they can see 1's and only 1's hat.
Anyway. Three logicians walk into a bar, and the server says "three beers, is it?" The first logician says "I don't know", and the second logician says "I don't know", and the third logician says "yes, please".
The warden is the real true logician.
Not so! The warden missed the possibility that the four prisoners are criminal logicians.
No! The logicians had a 50/50 chance of going free even if they just guessed. That's terrible wardening.
@@pauldzim actually, the chance is 2/3. because the second guy knows the guy in front has red, so there are 2 blue and 1 red left. And if you compound the fact that the chance of the 4 prisoners being logicians is > 0, it means that the chance that the prisoners would go free is actually more than 2/3
@@Simqerthe chance is in fact 1/2, as the first person can see 1 red and 1 blue hat which means there’s a 50/50 chance for person 1 to get correct
@@mynthics no, I am not talking about the chance for the left guy, I am talking about the overall chance for everyone to guess correctly. Nr 2 and 3 are a given, so 2 people will already have passed. The 50/50 is for the remaining 2
You made this so much more convoluted than it needed to be
That was... an unnecessarily long explanation. As other people pointed out, if they know there are 2 red hats and 2 blue hats (which is the case according to the sources in description), logician 1 will have to remain silent, as seeing different hat colors on 2 and 3 means he could have either color. Then logician 2 notices 1's silence, and thinks, "If 3 and I had the same color, 1 would immediately know his own hat color. As this was not the case, 3 and I have different hat colors, meaning that since 3 has a red hat, my own hat is blue".
Same reasoning. Glad I do not have to write all that !
i think it's not the first time he presents this puzzle either.
For bonus points, by the same logic 3 immediately knows his own hat color as soon as 2 calls out.
I don't think there's any way for 1 or 4 to know their own color, though.
One more of them could know, as at some point either 1 or 4 would have to guess their hat colour and right/wrong this would inform the other person.
im not sure how 3 would know their colour, as 3 can only figure out his hat is different colour to 2 but 3 cant see 2 to know the exact colour
Let's assume that "they are not allowed to talk to each other" means that they can't communicate to each other at all. Then the only way that #2 would know that #1 was uncertain was via the fact that the warden has not already set everyone free. The problem is therefore better posed if there are time constraints.
I think they'd notice being set free
This is what I thought if they can't talk to each other then how are they able to communicate halfway through??
@@ashleybroughton7713 Even if they couldn't hear each other answer, Logician 2 would still realize they haven't been set free after a few seconds and then figure out what's going on.
Even if that weren't the case, there's a second solution here. The logicians were given the information that the Warden specifically set up this scenario to test if they were logicians and not criminals. For this scenario to even be a logic puzzle, logicians 2 and 3 have to have different colored hats since any criminal in the 1 spot could figure out his hat color if he sees two of the same colored hats in front of him. With that information alone, Logician 2 should be able to instantly say he has a blue hat, even if he never gets any information from Logician 1.
"Let's assume that "they are not allowed to talk to each other" means that they can't communicate to each other at all."
- My point exactly.
Many of these logical problems seem to rely on ignoring the initial conditions, not actually working out the logic.
What a convoluted way to explain a simple solution.
I think firmly stating that they can't communicate in any way was a little confusing here. If we had clarified that there are clock ticks, and on each tick a logician can answer (and all other logicians take note of when no-one else answers), then it would've made it more clear that the passage of time could be used to communicate whether a logician does/doesn't know the answer. It's a pattern used in other hat style logic puzzles so it's not too far out of reach to imagine, but it's worth clarifying.
e.g.: the blue-eyed people on an island logic puzzle uses the concept of "days" to let the puzzler understand that the passage of time is divided into distinct rounds, which can tell agents at which point other agents were uncertain of the answer.
Exactly. I don't need the fluff story. Just clearly state the rules. I had to look it up somewhere else to understand that they get turns to answer or decline and it immediately became clear that the first logician not answering is the information the second one needs to know his hat is different color than the third.
At timestamp 0:46 the warden says they are not allowed to talk to each other. So right at 3:17 when #1 talks, the warden should have said in the voice of Bill Paxton from Aliens, "Game over, man", and they should have been marched back into jail.
Exactly. Every logician should only find out if another logician guessed right after taking a guess himself/herself. Or more generally, no communication, since silence is communication as well, it communicates that 2 and 3 have different hats. That'd make the problem unsolvable though, so the flaw had to be intentional.
I misheard the instructions at first and try figuring out how they could all state the color of the hat for certain. I’m glad I listened to the instructions again.
Surprisingly, 3 is the next person that can guess his hat, after 2 shouted blue then if 3 is blue, 1 would shout right away, so 3 knows his hat is red
3 does not even have to wait. He knows that he has different color than 2, for same reason 2 knows. So when 2 says his hat color, 3 immediately knows his color.
Then 1 and 4 have no way to determine their color, as no one sees them. Which is a bit disappointing.
Even better: If the distribution of hats is arbitrary, logicians 2 and 3 are nevertheless always sure to find out their hat colours. The other two only find out if they wear the same colour (and 2 and 3 wear the other colour). Plus, logician 3 doesn't even have to know the order of people behind him. He only has to wait for someone to call out a colour and will then be sure he wears the opposite colour.
@@yafriendceko That's the neat thing here, 3 doesn't need to know. All they know is that the first one to call out a colour (provided they only do so once they're 100% certain, which is generally assumed anyway) cannot wear the same colour as 3 does.
I got asked a version of this in an interview about 10 years ago. I remember when the question was posed to me I thought there couldn't possibly be an answer. But the interviewer asked me to try working it anyway. I spent an agonizing amount of time on it, building a logic table. Eventually I figured it out and was very happy! I didn't get the job, but I was still pleased with myself for answering this puzzle correctly in a nerve wracking situation.
I’ve interviewed a lot of people and I think question like these are MUCH harder in a stressed situation, so kudos to you
That question got asked in Honkai Star Rail too xD
In short:
After some waiting, 1 stays silent, meaning 1 and 2 aren't the same color (he'd know his color otherwise)
2 understands that now and knows his color must be blue, since 3 has red.
So 2 will be able to say his hat color with certainty
An easy puzzle after a long time.
But i feel that you unnecessarily elongated the solution by giving all possibilities.
Just say that 1st person didn't shout his colour which give information to 2nd one that his colour is different than 3rd one.
From the problem statement, it is not clear that the first person can see the next two hats. It is really important to know this information for problem solving
Your solution works perfectly, but it is unnecessarily complicated. The beginning with excluding 3 and 4 is fine. But then 1 and 2 can forget about any combinations. If 1 saw two blue hats, he'd know his hat is red, and vice versa if he saw two red hats, he'd know his hat is blue. Since he's not saying anything, 2 realizes 1 must see both colours on his and 3's head and concludes his hat is red if 3's is blue and blue if 3's is red.
I didn't iterate the possible arrangements but just worked out that in general, Logician 4 can see 2 logicians and if they have different colored hats, he cannot be certain of the color of his own hat. So, given Logician 1's uncertainty, Logician 2 knows his hat must be the opposite color to that which he can see on Logician 3, hence he announces he is wearing a blue hat.
Logician 4 doesn't see any other, just like logician 3...
@@Azarathification Yeah, I realized after that when I said Logician 4, I meant 1 but now this is getting complicated to explain.
You should include a timeframe for when they answer to make the puzzle more clear something like every 5 minutes the warden will ask for an answer so that the time between isn’t arbitrary
5 seconds would be enough for person 1 to realize that person 2 and 3 have different colors. After those 5 seconds, person 2 can say that his color is the opposite of the one in front of him.
@@Bob94390 true it may be enough but if you’re a perfect logician and you aren’t given a timeframe for answering and you know you have to wait for the second answer slot how do you know how long it will take for the first answer slot to go unanswered. Maybe the person in spot 1 is also a perfect logician but they take a little more time than you think they would and end up with a wrong answer. The time slots make the puzzle cleaner for us and the fake computer people the puzzle is interested in.
Any logicians should be able to figure out “I see both red hats so I must be wearing blue” almost immediately. Unless Person 1 is daydreaming or something, it shouldn’t take more than a few seconds for Person 2 to be certain that Person 1 doesn’t have enough information to know. If anything, you just need to add the assumption that everyone’s actually paying attention.
No time-frame needed. No. 1 would know the answer instantly (sees two hats of same colour) or never.
By your reasoning, we have to also account for the fact that no.2 could be an a**hole who wants to be in prison and never speaks up (although he knows the answer).
@@DreadX10 in this configuration yes but what if it’s a different configuration and no1 takes an extra second before answering. No2 has suddenly trapped them all in jail. The whole point of these puzzles is that they’re guaranteed with perfect logic but not giving them a set time to answer can lead to mistakes. If no2 solved it slightly faster than no1 and then answered and the configuration was different that would be it. With an answering cadence problem solved.
Since they're not allowed to communicate, this timing strategy does also kind of rely on chance. My take was that the logician 1 must say "I'm 50% certain that...".
Now solve for the amount of time #2 must wait before deciding that #1, doesn't know.
An interesting variation of a this kind of problem is this: 3 logicians wanted to figure out which of them is smartest, so they went to the king. The king devised a test in which they were to sit in a circle and close their eyes. They were told that a hat, either red or blue, will be placed on each of them. They were also told that there are 2 blue and 3 red hats in total. When they open their eyes, they are supposed to figure out which hat colour they are wearing. Each will only see the hats of the two logicians in front of them. The first to shout out their hat colour is the smartest. While their eyes were closed, the king placed a red hat on each of them. When they opened their eyes, after a short time, one of them shouted that they were wearing a red hat. He was declared the smartest. The question is, how did he think?
Not seen that variation, but it's cute.
1. If any of them were looking at two blue hats, they would instantly know they were wearing a red hat, but no-one answers so it's not two blue hats.
2. Having figured this out, if someone saw one blue hat, they would instantly know their hat was red, but no-one answers meaning there isn't one blue hat.
3. The only remaining possibility is all 3 hats are red, and so it's just down to who is fastest to answer red.
@@manudude02 The logic is nice, but you start to get a difficulty with how to define 'instantly + instantly'. How long should one of them wait to determine that they are now in the second consecutive silence and not the first one? Because of this difficulty they cannot simply rush to be the fastest to say 'red' the second they reason by this method that it could logically be the answer, just in case one or both the others are still in the first of the two consecutive silences.
The king asks the first time and silence, so it excludes 2 blue hats.
The king asks second time and silence again, so it excludes one blue hat.
When the king asks for the third time, they all know they're wearing red hats, because that's the only remaining possibility, and the fastest to speak wins.
The test needs to be fair, so all of them must be wearing the red hat. Therefore the first to realise that answered the question. I know this doesn't employ logic but common sense but I feel the given task isn't suited for it
@@manudude02there were 2 blue hats😐 so if one blue hat was in front of me then how would I assume I m wearing red hat? It can still be blue hat?
you've brought in a time dimension into the problem, and also the rule of not talking to one another is broken if silence is considered communication
Oh Presh, you do love an overcomplicated solution. This takes exactly two very easy-to-follow steps-no formulae, no tables. • If #1 saw the same color on #2 and #3, he would shout that he had the other color. • He doesn't shout, so #2 knows he is wearing a different color than #3 and shouts out his hat color. DONE. Why would you start with the two who have least information?!
Interesting, heard this logic problem on a car talk radio program many years ago. Like a good humorous story nice to see this make the rounds again. Thanks for sharing.
Simple explanation (spoilers!):
If L1 saw two hats of the same color, he'd know that his hat was the other color. Since L1 does not know his hat color, L2 knows that his own hat must be the color that he does not see on L3.
The flaw is that 2 can logically never be sure that 1 isn't going to speak up. He can't know how long it is reasonable to wait before assuming 1 isn't going to speak up - that element is social, not logical, as I'm sure any autistic person will agree. If the warden explicitly asked each person one at a time then this would be resolved.
Number 1 remains silent since he sees 1 blue and 1 red hat, leaving two options open. Number 2 (logician as he is) realizes that when number 1 remains silent, number 1 must face 1 blue and 1 red hat. Since number 2 can see number 3 is wearing a red hat, he knows for sure he is wearing blue himself.
Understood bitzh
except when it comes down to number 1 and number 4 still not knowing which color hat they are wearing.
I've seen one similar to this on TED-Ed, though that was with more people, and a random distribution of hats. In this case, if the top person sees two red hats, he knows he has blue, and thus they win, same thing if he sees two blue hats, he knows he has red. However, if he sees a red and a blue, he has a 50/50 chance, and thus hesitates to answer. This then tells the next person in line that he sees a hat of each color, since he doesn't automatically know his own hat color. So he looks at the hat in front of him, and instantly knows his own hat color, being the opposite of the person in front of him, thus they win. The third and fourth people don't even need to answer.
Hmmm but you said they can't communicate with each other at all. Person 2 can only guess their own hat is blue because they were awaiting for the shout of Person 1 and didn't hear it, that is a form of communication. If the one who knew their hat color had to whisper it to the warden instead of shouting, there would be no solution, and no communication like the rules stated.
I’ll go one further. Even with the stipulation that this counts as communication (which it absolutely does, you nailed it), the puzzle is trivially soluble. The illustration shows that the hats are brimmed, and the narration makes no claim that the men cannot see their own hats.
So, an acceptable answer is: All 4 men can immediately state with certainty the color of their own hat, because they can see their own hats.
@@jaarnealahah fair enough! guess you could say they just had to _look up_ the answer
No good. They may only look ahead, not up. And no cheating by whispering either. Only pure logic.@@zewzit
This logic was quite simple. I didn't use your method exactly but somewhat similar to your answer. As logician 1 is confused, means 2 will understand that the hat of 2 and 3 are of different color. So he sees 3 has red color and concludes 2 himself is blue.
The riddle can be solved by just Logician 1, 2 and 3 interacting with each other. Logician 4 is pretty much just a distraction for the solver. It would be the same riddle with 3 guys, 3 hats, where 2 is the same color and one is different.
Solved it in under a minute, but, then again, I had "help"; I already knew a similar problem (the Three Chinese Philosophers riddle, in which a queen paints either a red or a green dot on each of three philosophers' foreheads and challenges them to find out, without conferring, whether their own dot is red or green). To be perfectly honest, I thought Presh's explanation was a bit over-complicated; he could have been a bit more succinct and to the point.
I seem to remember a harder version of this puzzle where all 4 men had to guess the color of their hats. As soon as 2 guesses, number 3 immediately knows the color of his hat as well. I don't remember how 1 and 4 got their color but I think it had to do with the wording of the puzzle.
Yeah, I was trying to figure out the same. Like, what'd be different if 1 and 4 had their colors switched?
@@couchpotatoe91nothing because in order for them all to get out only one person has to know their hat color
@@drewsworthh ah ok. I remember a prisoner scenario where everyone has to guess their color in order to be freed. But it had some twist iirc like the guy behind the wall was actually on the very left behind another wall and could only glimpse the tip of the last person's hat.
The difficult version is 100 people (could have been any number) lining up. There are 2 colours but the proportion of hats is not given. Everyone takes turn guessing starting from the back. They're allowed time to strategize. They can only make a guess but not signalling using other things like volume. The optimal strategy ensures everyone except the one at the back can deduce the colour correctly.
Simple answer:
- Logician 2 knows that Logician 1 can see 2 and 3.
- And therefore if 2 and 3 share the same color, Logician 1 will know his color.
- Since Logician 1 doesn’t say anything, this only means 2 and 3 have different color.
- Logician 2 can see red on Logician 3 ==> he has a blue hat
I've always enjoyed puzzles, particularly logic puzzles. I first heard this one around 25 years ago from a guy who worked in the carpark of the company where I worked. When he told me, I thought about it for a few minutes before realising the answer. As I told him he stated that he already knew the answer but didn't know the reason why, which I was able to explain.
This is a very easy puzzle. But it was explained the very complicated way. That’s kinda the opposite of what was supposed to happen
This was a very very easy puzzle. I’ve never solved one of these before, but this one was extremely easy
You give the rules that they cannot communicate to each other, but then says #2 knows their hat because they have the information that #1 see 2 different colors. But without communicating he cannot know that …
Excellent video! I went through this a few times to fully understand the logic (ok, I'm slow), but now I do and I think this is terrific!
You don't really need the first step (2 eliminating half the possibilities). 1's silence immediately means that 2 and 3 have different colours, so upon noting 1's silence 2 can announce his colour as the opposite of the one he sees on 3. The question, though, is how long should 2 wait to be sure that 1 doesn't see two hats the same colour rather than, say, is still fumbling with his glasses?
Good catch
The obvious answer is to look at the brim of your own hat
😂
Should have more likes. XD I saw this puzzle before with simple conical hats, but didn't think about why they need to be, haha.
What I find interesting is we use a combination formula, 4C2, to determine a permutation ( the ordered number of ways they can be arranged with the hats).
haha best comment!
I know a different yet similar logic puzzle:
Three Logicians are behind another. They each have a red or blue hat. Logician 1 can see both other hats, Logician 2 can see one and Logician 3 can see no hat. They are told, that the capturer has 2 red hats and 3 blue hats. The not used hats were hidden. After a long while Logician 3 says with certainty his hat-color. Why and which color is it?
If Logician 1 could see two red hats, he could assume, that he has a blue hat. But as he says nothing, he can't see two red hats.
Therefore, if Logician 2 could see a red hat, he could assume, he has a blue hat, because otherwise Logician 1 would say his color.
Therefore, Logician 2 can't see a red hat, so Logician 3 knows, that he has a blue hat.
I overheard the fact that only one prisoner needs to answer correctly.. So I tried to solve this for like ages until I watched again. But then I thought the solution is to easy if 1 can see 2 and 3. What I'm trying to say is that I spend an unreasonable amount of time trying to solve this puzzle.😂
I love "a group of logicians" problems. I thought your solution was overly complex though -- you simply have to realize that when 1 doesn't speak, 2 realizes that means that he and 3 have different colored hats, and since 3 has a red hat, his must be blue. (If 2 and 3 had the same color, 1 would know their hat color immediately).
If hats on 2 and 3 are the same color 2 can expect 1 to speak, but it 1 does not speak for a time, 2 can assume that his hat is of the opposite color to 1 whose hat he can see and can speak.
Pretty simple actually. The first man would speak if he saw two identical hats in front of him. Since he doesn't, it should become clear to the others after a few minutes that positions 2 and 3 contain one red and one blue hat. The second man would then know this, see a red hat in front of him, and state that his hat is blue.
I used to present this to my undergrad students years ago, plus three different cute variations of this riddle (three guys in a circle seeing each other, then once again but one of them blind, then many logicians on a train)… Will you present those advanced versions too, in another video? 👍
The assumption is that all four prisoners know how many of each hat is in play and that each credits the other with being perfectly capable and infallible logicians.
One can think of the problem as having rounds. Each prisoner enters his response on each round. After each round all the prisoners know each of the other prisoners' responses. The responses are PASS, RED, BLUE.
Number 2 can see that number 3's hat is red. If his own (number 2's) hat were red, then number 1 would instantly know that his hat is blue. Number 1's silence convinces number 2 that his own had must be blue. No one else can tell what color their own hat is. Until number 2 speaks, number 3 knows for sure that he and number 2 have different colors.
On round one all prisoners pass.
On round two, number 2 can infer from number 1's pass that he, prisoner 2 has a blue hat, so the responses are:
PASS, BLUE, PASS, PASS
On round three:
RED, BLUE, PASS, PASS.
No further progress is possible, but at this point 1 and 2 know all four colors, and everyone knows 1 and 2.
how you commented 2 days ago video is published few minutes ago only????????
@@uditisgaming5872 What are you, a logician?😂
No, it's not an assumption.
The 4 are logicians and friends, so they know this info.
And the Warden told them it's 2 red hats and 2 blue hats, so all is known, no assumptions required.
Actually, round three (assuming the answers are not given aloud in order) would be: pass, blue, red, pass. After Number 2 is able to deduce that he is wearing a blue hat, Number 3 knows that Number 1 saw a blue hat on Number 2 and a red hat on him (Number 3). If each person responds aloud and in order of their number, then Number 3 would announce the color of his hat in round 2.
I agree with the idea but the results are wrong
Round 1: Pass, Pass, Pass, Pass
Round 2: Pass, Blue, Pass, Pass
Round 3: Pass, Blue, Red, Pass
That's it, it's not possible for person 1 and 4 to guess their colors so it's 50/50 chance.
You made that explanation so much more complicated than it needed to be! All you needed to say was that there were only two hats of each color, and thus if logician one saw two hats of the same color he would know that he was looking at all the hats of that color and thus his hat must be the other color, and logician 2knows this, so if he doesn’t hear logician one state the color of his hat, then he and logician three must be wearing two different color hats and thus his hat is the color he *doesnt* see in font of him. (And fwiw, once logician three hears logician two shout out that he’s wearing a blue hat, he would know that his own hat must be red, not that he would have any reason to care at that point.)
If 2 and 3 had the same color, 1 would announce they’re wearing the opposite color. 2 could hear that announcement but it isn’t made, despite allowing one whole second for a perfect logician to make it. 2 then knows that they’re wearing the opposite of 3, i.e. blue. 4 is a little annoyed to be a background prop just to wear an unseen hat, but is glad to go free.
I find these logic problems so ridiculous sometimes. Why take a simple thing and make a math problem out of it? Thats madness😅
Your channel is awesome but this is the worst explanation of a problem ever. Much more simply stated as “If position 1 sees two hats the same, he calls out other colour, but if silent, position 2 calls out the opposite of what he sees 3 wearing”.
You must not have had many things explained to you if you think this explanation that takes one unnecessary detour is "the worst explanation of a problem ever"
@@noahblack914its still a lot longer than needed, we didnt really need math or anything, this comment is way easier to explain and to visualise
I really love these logic puzzles.
these guys cannot communicate with eachother.
Proceeds to make them communicate.
Plot twist: Logician 4 is watching them solve this with the warden
Excellent, I was searching this puzzle since long.
_"After a while after #2 doesn't answer"_ is kind of a dishonest modification to the rules.
m1 - nice
m2 - well done
m3 - good work
m4 - don't even have to be here
autor - slowpoke
another variant of this question that teacher asked us at school
some1 have 5 hats (2 blue and 3 red) and he will put 3 of them on 3 ppl and place them in circle in front of each other, if some1 says his hat color they win, other rules is the same
This is so incredibly easy. It was so easy in fact, that I questioned if it really only took more 4 seconds to figure out.
Just like the honesty of thumbing up your own comment.
This explanation is way more complicated than it needs to be.
The quickest way to describe the solution is the following.
Logician #2 sees a red hat in front of him. He knows that if his hat was red that #1 would see two red hats and #1 would immediately deduce that his hat was blue, because there can only be 2 red hats ( which didn't happen ). Thus, #2 is able to definitely deduce that he is wearing a blue hat.
If I was in #2's positions, I'm definitely not considering all the position configurations. I'm just thinking of what would happen if the guy behind my saw 2 red hats.
Must admit, I didn't even think about the probability. Just that 2 would know that 1's inability to determine what he's wearing must mean that he's seeing a blue hat and a red hat and the fact that 2 can see a red hat means he must be wearing a blue hat. Thanks for posting as I can now explain to my wife that I am fully entitled to a celebratory glass of wine.
Idk if I would say this is strictly a logic puzzle, since the crucial step in it is an inductive inference.
1 taking a long time to answer does not strictly entail that he is stumped, since he could have for example, just fallen asleep, or struck a deal with one of the guards, or maybe has a grudge against the other logicians. However improbable these scenarios might be, they prevent this from being a logic puzzle strictly, since it relies on the assumption that the first logician would take a long time to answer only if he didn’t have enough information to complete the puzzle.
Still though a very interesting puzzle, and enjoyable video
This puzzle proves, that studying formal logic is best preparation for criminal career.
That's an overcomplicated solution.
You don't need possibilites.
If #1 sees 2 same color hats on #2 and # 3's head, he can tell his hat's color.
If he doesn't tell anything, #2 knows that he and #3 wear different color hats. Because he sees a red hat he knows he wears a blue one.
What throws people off is that they think that No. 1 must be the one to answer as he has _the most information_ ...!
I think the core of this problem is that while they cannot speak to each other, they are able to communicate through silence
1: #3 and #4 do have information. They have the same information about the lack of response from #1 that #2 has.
2: Nobody said they had give a true or specific response. They just had to respond with certainty. Any one of them could say "My hat is certainly red or blue!" or even "My hat is certainly green!" and win freedom for all of them.
Oh that is great. I love that that "time" is the "hidden" axis here.
I was so relieved I got this, for some reason.
I can write it simpler. If #1 saw two hats of the same color, he would instantly know his is different and shout that out. Since he didn’t #2 can conclude that his hat is DIFFERENT from #3’s hat, so he can shout out whatever color is not on #3’s hat.
Incredible explaination!
I got as far as working out the probabilities and how it would be simple if 2 and 3 had the same colour. I didn't think of the final step. Very clever.
This is a simple problem of pure logic, and by that I mean SIMPLE, and LOGIC. There's no need to introduce any mathematics whatsoever. Sometimes you really can reason without formulas.
Its interesting that this problem is presented with one specific permutation of hats since any permutation of hats will allow the prisoners to escape.
Case 1: L2 and L3 have the same hat color. L1 answers instantly of his hat color as it must be different than the 2 hats he sees.
Case 2: L2 and L3 have different hat colors. L1 does not answer since there are 2 possible permutations from what he sees. L2 reasons that L1 would answer instantly if this were Case 1, thus L2 knows they have a different hat color than L3 and so L2 answers correctly.
What an overcomplicated explanation for such a simple solution! I'm neither a logician nor a mathematician, yet I figured it out in less time than it took to explain it.
if the hats were all one color throughout the whole hat, all they'd have to do is look up at the brim of the hat to see what color they are wearing.
It's a nice puzzle, took me a few minutes to figure it out. But it's simpler to explain without any Math. Person 2 notices that person 1 doesn't say anything, which can ONLY be the case if person 2 and 3 each have a red and blue hat. Would they have both red or blue hats, person 1 would immediately know his own and say it. Given that person 2 sees person 3's hat being red, the only way is that his own hat must be blue.
One of my all time favourites, this. I first heard this during a 7-hour bus journey on a school trip when I was 10, and I have been confusing and baffling friends with this for 16 years ever since. VERY few people I have done this on have been able to work it out. They are all convinced there is no answer to this, and there has to be some sort of cheating involved. The stories people have come up with that involve cheating! (Reflective watches! Really!!)
I think the difficulty is, everyone assumes logician 1 will get the answer because they have the most information (they can see 2 hats). And they discard the possibility that it is in fact one of the ones with less information that can get it. So they try to come up with all the ways logician 1 could get it... and then promptly get stuck.
This is an interesting and intriguing logic problem. It is old school from the 60's if not before.
2:20 you're overthinking it. You don't need combinatorics or tables. If 1 sees two red or two blue he'll know his own color with a couple seconds' reflection. Once 2 has given him that time, and not heard an answer, 2 knows he differs from 3. MORE GENERALLY, 2 knows that his hat plus everything he sees before him won't yield an answer. As long as the puzzle would have let 1 solve the problem had 2 one color, 2 knows he has the other color. For instance: 10,000 people are on the hill, with the hats split 50/50. 1 sees 4999/4999; if he saw 5000 of either he'd know his own hat. 2 sees 4998/4999 and knows his own hat is the 4998 color.
Instead of 6 groups you can simplify to two groups of outcomes: that either the two hats in the middle match, or they are different. If they match number 1 says the opposite colour to the 2 he can see. If he stays silent then 2 says the opposite colour to the one in front of them.
I was given a very similar logic problem as a kid.
In this problem, there are 3 logical prisoners. The warden makes them sit in a triangle formation, all facing each other. He tells then he will place either a red or a blue dot on each of their heads. He tells them if they can deduce the color of the dot on their head, they will be set free, but if they guess wrong they will be executed. He also tells them they must raise their hand if they can see a red dot on anyone else, and if they fail to do so they will be executed. He tells them they will be given a 30 minute time limit to come up with a answer, and if no one answers they will play the game again, this time with each person being randomly assigned a new dot color. They are told they will play until somebody wins or loses. In the first round, the warden puts a red dot on each prisoner's forehead. All the prisoners raise their hands, because they can see a red dot. After about 15 tense minutes, one prisoner confidently announces, "I have a red dot" is set free. Why was he so sure he had a red dot?
Another prisoner looked at the 3rd guy with a blue dot for 15 minutes.
The stated solution hinges on breaking the rules - #1 has to talk for #2 to figure it out, and the rules say no talking.
However, #2 can merely take #1's silence to know that he is not wearing the same color hat as #3.
Yeah that's exactly what he's saying, he never said anywhere in the video that #1 tells #2 that he's confused.
@@joshuasgameplays9850 he literally says that #1 states that he can't deduce what hat he is wearing.
@@izzmus When does he say that?
Why complicate with all the possibilities? If Logician 1 sees two of the same color, s/he will speak out immediately. When that doesn't happen, Logician 2 knows his/her/their hat is the opposite color of Logician 3.
Thank for you this one. really enjoyed it
It surely was a very fun game to play for logicians 3 and 4!
I was about to post the exact same thing as @londonbobby. The explanation given in the video was unnecessarily complex. The easy explanation is: If #1 sees two hats the same color, then he speaks out immediately. If he doesn't speak out immediately, it means #2 and #3 are different, and this information is communicated by #1 being silent. So #2 knows the color of #3, and #2 knows his hat must be different from #3, so #2 speaks out.
Congrats. You basically regurgitated exactly what the video said but left out 75% of the actual explaining part. 😁
I love how logician 3 and 4 are just looking at a wall doing absolutely nothing even though they might know the logic of the puzzle.
When I first saw this logic puzzle in the late 70’s, the three prisoners (Never called logicians in the original.) were on the same level, 1, could only see 2’s hat, not 3’s. How is that older and much harder version worked out
Person 2, upon realising that person 1 doesn't immediately know what colour their hat is, can deduce that the colour of their hat is not the colour of person 3's hat
Otherwise person 1 says the colour of their hat with certainty
Nice video. Good channel. Keep up the good work.
There is another similar puzzle. There are three prisoners. And there are 5 hats (3 black and 2 white). We are using only 3 hats. The prisoners know that there are 5 hats, but they don't know which 3 hats are being used. All the rules are same. Which prisoner will guess his hat? There are multiple answers according to the hat placement.
This is a classic. Cool to see it on CZcams
Logician 2 is the one who can say for sure what the color of his hat is. He sees the no. 3 and knows his hat is red. He knows that if his hat was red the no.1 would immediatelly yell the answer. Now that he is silent too, logician 2 knows no. 1 has the amount of information that makes him to guess on a 50/50 state. This implies that the other red hat is for sure for either no.1 or no.4 so the logician 2 yells mine is blue for sure!